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# Advanced Biostatistics Seminar 171 290

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This 64 page Class Notes was uploaded by Dedric Ritchie on Friday October 23, 2015. The Class Notes belongs to 171 290 at University of Iowa taught by Joseph Cavanaugh in Fall. Since its upload, it has received 32 views. For similar materials see /class/228028/171-290-university-of-iowa in Biostatistics at University of Iowa.

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mam midi awng lg wav wgiikw Lru1riiunenc iMl l ll li i i ml Leeime Kill I he Amglkg m Co lMl dlel Sellame thtem iza Joe Cava naugh Department of Biostatistics College of Public Health The University of Iowa May 3 2005 Joe Cavanaugh j W ngmm s imain Lect re Xl of Model Selection Criteria 9 Facts and Fallacies Concerning Model Selection Criteria 0 The quality of the fitted model selected by a criterion Data dredging and selection criteria Relative values of selection criteria Nested versus nonnested models The use of likelihoodbased criteria to compare models with likelihoods of different distributional forms 0 Hypothesis testing versus model selection criteria Joe Cavanau 29 Advanced 39 39 39 Seminar Model Selection The Appli Lion of Model Selection Criteria Facts and Fallacies 0 Fact or Fallacy The fitted model selected by a model selection criterion is necessarily a useful model 0 The preceding is a fallacy o In general a model selection criterion attempts to identify the optimal fitted model in a candidate family 0 However in some instances all of the fitted models in the candidate family will be inadequate the selection criterion may then be choosing the best of the worst Joe Cavanaugh 17129 Advanced Bio 39 Model Selection The Application of Model Selection Criteria Facts and Fallacies 0 Whenever possible interpretable measures that assess overall conformity of the fitted model to the data such as R2 in linear regression modeling andor propriety of structural fit such as the deviance in generalized linear modeling should be used to evaluate fitted models favored by a selection criterion 0 Burnham and Anderson 2002 p 62 AIC is useful in selecting the best model in a set however if all the models are very poor AIC will still select the one estimated to be the best but even that relatively best model might be poor in an absolute sense Thus every effort must be made to ensure that the set of models is well founded Joe Cavanaugh 17129 Advanced BioslaLi cs Seminar Model Selection The Appli Lion of Model Selection Criteria Facts and Fallacies 0 Fact or Fallacy The use of model selection criteria permits data dredging post hoc analyses involving the exploration of a large set of conceivable variables and effects The preceding is a fallacy with a footnote Post hoc exploratory analyses are inherent to statistical modeling yet an exhaustive consideration of variables and effects often amounts to a search for statistical significance and results in fitted models that are excessively tailored to the data at hand Reliance on model selection criteria provides protection from chasing statistical significance yet may still lead to fitted models plagued by spurious unimportant effects Joe Cavanaugh 17129 Advanced Bio 39 Model Selection The Application of Model Selection Criteria Facts and Fallacies Burnham and Anderson 2002 p 147 discuss an elk study by Cook et al 2001 where analyses were conducted using stepwise linear regression AIC and Mallows39 Cp Cook at al 2001 found that their approach produced a model that was biologically unreasonable unstable due to multicollinearity and overparamerized Burnham and Anderson 2002 p 147 claim that a common mistake in such studies is the failure to posit a small set of a priori models each representing a plausible research hypothesis Whenever possible the candidate family should be carefully configured based on a thorough consideration of the scientific principles governing the underlying phenomenon Joe Cavanaugh 17129 Advanced BioslaLi cs Seminar Model Selection The Application of Model Selection Criteria Facts and Fallacies Fact or Fallacy In a model selection application the optimal fitted model is identified by the minimum or maximum value of the criterion relative differences between criterion values are unimportant The preceding is a fallacy In general a model selection criterion scores every fitted model in a candidate family in accordance with how well each model balances the competing objectives of conformity to the data and parsimony The sizes of the scores are important models with similar criterion values should receive the same ranking in assessing criterion preferences Model Selection ll Advanced Bio aLi 39 The Application of Model Selection Criteria Facts and Fallacies a Question What constitutes a substantial difference in criterion values 0 For AIC Burnham and Anderson 2002 p 70 feature the following table AIC 7 AICmin Level of Empirical Support for Model 139 o O 2 Substantial 4 7 Considerany Less gt 10 Essentially None 17129 Advanced BioslaLi cs Seminar Model Selection Joe Cavanaugh The Application of Model Selection Criteria Facts and Fallacies o For SIC Kass and Raftery 1995 p 777 feature the following table slightly revised for conformity of presentation SIC 7 SICrmn Evidence Against Model 139 O 2 Not worth more than a bare mention 0 2 6 Positive 6 10 Strong gt 10 Very Strong Joe Cavanaugh 17129 Advanced BioslaLi cs Seminar Model Selection The Application of Model Selection Criteria Facts and Fallacies Fact or Fallacy A model selection criterion cannot be used to compare nonnested models The preceding is a fallacy Model selection criteria can and should be used to compare nonnested models However when nonnested models are compared using selection criteria the distributions of differences in criterion values are less straightforward to characterize Thus when comparing nonnested models using model selection criteria marginal differences should be considered carefully ll Advanced Bio aLi 39 Model Selection The Appli Lion of Model Selection Criteria Facts and Fallacies Useful quotes on the use of AlC type criteria 0 Burnham and Anderson 2002 p 88 A substantial advantage in using information theoretic criteria is that they are valid for nonnested models Of course traditional likelihood ratio tests are defined only for nested models and this represents another substantial limitation in the use of hypothesis testing in model selection Joe Cavanaugh 17129 Advanced Bio 39 Model Selection The Appli Lion of Model Selection Criteria Facts and Fallacies o Kitagawa 1987 states It must be emphasized that conceptually there are no difficulties in the comparison of Kullback Leibler information numbers of nonnested models and there is a significant difference in the concept of likelihood as an estimate of the Kullback Leibler number from the one in the statistical testing framework However Kitagawa 1987 adds differences in AIC values may be more variable in the nonnested situation But here again although some care is required in comparing marginal differences there are no conceptual difficulties in comparing nonnested models Joe Cavanaugh 17129 Advanced Bio 39 Model Selection The Application of Model Selection Criteria Facts and Fallacies Fact or Fallacy AIC and other likelihood based criteria eg TIC SIC HQ cannot be used to compare models that have likelihoods of different distributional forms eg normal Poisson binomial The preceding is a fallacy Model selection criteria can be used to compare models that have likelihoods from different parametric families The validity of such comparisons however is not supported by many of the theoretical criterion justifications that have appeared in the literature 17129 Advanced BioslaLi cs Seminar Model Selection Joe Cavanaugh The App on of Model Selection Criteria Facts and Fallacies 0 When the criteria are computed no constants should be discarded from the goodness of fit term 72 In fyl k such as n In 27139 3 Keep in mind that certain statistical software packages routinely discard constants in the evaluation of likelihood based selection criteria eg in normal linear regression n In 62 is often used as the goodness of fit term as opposed to n In 82 nn 27139 1 Joe Cavanaugh 17129 Advanced B Seminar Model Selection The App n of Model Selection Cr Facts and Fallacies Illustrative AlC application Burnham and Anderson 2002 673 feature an example where the response variable is T4 cell counts per cubic milliliter of blood The response is measured for 20 patients in remission from Hodgkin39s disease and for 20 controls Several probability distributions are considered for the response including normal log normal gamma Poisson and negative binomial Two mean structures are entertained one that allows for two means for the two different groups ie allows for a treatment effect and one that does not Joe Cavanau 29 Advanced Seminar Model Selection The Application of Model Selection Criteria Facts and Fallacies Results One Mean Two Means Distribution k AIC k AIC Normal 2 6088 3 6064 Log Normal 2 5901 3 5886 Gamma 2 5913 3 5880 Poisson 1 116520 2 102040 Negative Binomial 2 5892 3 5860 Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection The Appli Lion of Model Selection Criteria Facts and Fallacies Conclusions 0 There appears to be evidence in favor of a treatment effect for each distribution the AIC value for the two mean model is less than the value for the onemean model a The negative binomial distribution appears to be favored slightly by AIC o The Poisson distribution appears entirely inappropriate for this data 0 The Poisson models are the only models that do not include a separate dispersion parameter Joe Cavanaugh 17129 Advanced Bio 39 Model Selection The Application of Model Selection Criteria Facts and Fallacies 0 Fact or Fallacy Hypothesis testing offers a superior paradigm for model selection to model selection criteria 0 This point is controversial although many would deem the preceding a fallacy o Regardless in considering hypothesis testing for model selection one should keep several important points in mind ll Advanced Bio aLi 39 39 Model Selection The Appli Lion of Model Selection Criteria Facts and Fallacies a First classical hypothesis testing procedures are generally based on the assumption of nested models with the larger model represented under the alternative hypothesis being regarded as correct o In many statistical modeling applications especially those in the biological and ecological sciences the notion of any model being correct is difficult to defend Although the derivation of certain model selection criteria such as AIC involves the assumption that the true model is a member of the candidate family selection criteria can be effectively used in settings where this assumption is tenuous Joe Cavanaugh 17129 Advanced Bio 39 Model Selection The Appli Lion of Model Selection Criteria Facts and Fallacies o In hypothesis testing however the interpretation of p values power and statistical significance relies upon the notion of one or the other hypothesis representing truth 9 CR Rao and Y Yu 2001 Since in practical situations the assumed null hypotheses are only approximations and they are almost always different from reality the choice of the loss function in test theory makes its practical applicability ogicay contradictory Joe Cavanaugh 17129 Advanced Bio 39 Model Selection The Appli Lion of Model Selection Criteria Facts and Fallacies 0 Second the p vaue depends on both the effect size and the sample size In large sample settings even unimportant effects may be deemed statistically significant 3 IR Savage 1957 Null hypotheses of no difference are usually known to be false before the data are collected when they are their rejection or acceptance simpy reflects the size of the sample and the power of the test and is not a contribution to science Joe Cavanaugh 17129 Advanced Bio 39 Model Selection The Appli Lion of Model Selection Criteria Facts and Fallacies 0 Third as a general rule hypothesis testing has greater relevance in experimental settings than in observational settings 9 Burnham and Anderson 2002 p 83 A priori testing plays an important role when a formal experiment ie treatment and control groups being formally contrasted in a replicated design with random assignment has been done and specific a priori alternative hypotheses have been identified Model Selection 17129 Advanced Bio Joe Cavanaugh will wing laikgra wgiikw Sgiiulriiimelrc ll rl li ll S l l l l Vi The Imifmmg i m Qn mrim fill J Joe Cava naugh Department of Biostatistics College of Public Health The University of Iowa February 22 2005 Joe Cavanaugh mmgmmms witnein J l v IC D tiun D Lecture VI The Schwarz Bayesian Information Criterion SIC 0 Overview of SIC o Derivation of SIC 0 Discussion SIC and Bayes Factors ll Advanced Bio aLi 39 39 Model Selection Overview SIC Derivaiiun Overview of SIC Key Constructs 0 True or generating model gy 00 o Candidate or approximating model fyl0k a Candidate class Hk fYl0k Wk 6 900 o Fitted model fyl k Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Overview Evil Derivniiun Overview of SIC o Akaike information criterion AIC 72 In fyl k 2k 0 Schwarz information criterion SIC 72 In fyl k kln n o AlC and SIC feature the same goodnessof fit term 9 The penalty term of SIC is more stringent than the penalty term of AIC For n 2 8 kln n exceeds 2k 0 Consequently SlC tends to favor smaller models than AIC Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Overview Overview of SIC o The Schwarz information criterion is often called the Bayesian information criterion 0 Common acronyms SIC BIC SBC SC AIC provides an asymptotically unbiased estimator of the expected Kullback discrepancy between the generating model and the fitted approximating model a SIC provides a largesample estimator of a transformation of the Bayesian posterior probability associated with the approximating model By choosing the fitted candidate model corresponding to the minimum value of SIC one is attempting to select the candidate model corresponding to the highest Bayesian posterior probability Joe Cavanaugh 17129 Advanced Bioslali cs Seminar Model Selection Overview Evil Derivatiun D Overview of SIC o SIC wasjustified by Schwarz 1978 for the case of independent identically distributed observations and linear models under the assumption that the likelihood is from the regular exponential family a Generalizations of Schwarz39s derivation are presented by Stone 1979 Leonard 1982 Kashyap 1982 Haughton 1988 and Cavanaugh and Neath 1999 0 We will consider a justification which is general yet informal Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Derivation of SIC 0 Let y denote the observed data 0 Assume that y is to be described using a model Mk selected from a set of candidate models Mk1 Mk2 MkL 0 Assume that each Mk is uniquely parameterized by a vector 0k where 0k is an element of the parameter space k k E k17 k2 7 0 Let L0kly denote the likelihood for y based on Mk 3 Note L0kly fyl 0k 0 Let k denote the maximum likelihood estimate of 0k obtained by maximizing L0kly over k Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection v SIC Derivation D Derivation of SIC 0 We assume that derivatives of L09lt l y up to order two exist with respect to 0k and are continuous and suitably bounded for all 0k 6 k 0 The motivation behind SIC can be seen through a Bayesian development of the model selection problem a Let 7rk k E k17 k2 kL denote a discrete prior over the models Mk1 Mk2 MM 0 Let gt9lt l k denote a prior on 0k given the model Mk k E k17 k2 7 Joe Cavanaugh 17129 Advanced Bioslali cs Seminar Model Selection J SIC Derivation D Derivation of SIC 0 Applying Bayes39 Theorem thejoint posterior of Mk and 0k can be written as 7rk 09k l k L0k l h k7 9 y where my denotes the marginal distribution of y o A Bayesian model selection rule might aim to choose the model Mk which is a posteriori most probable o The posterior probability for Mk is given by Fwy my1wk m M gwklk dok Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Derivation of SIC 0 Now consider minimizing 72 In Pkly as opposed to maximizing Pkly 0 We have 72 In Pk ly 2 In my 7 2 In 7rk 72m Liokiy gwkik cm 0 The term involving my is constant with respect to k thus for the purpose of model selection this term can be discarded Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection J SIC Derivation D Derivation of SIC 0 We obtain 72 In Pkly X 72 In 7rk 2InL0kly gwklk cm E kly 0 Now consider the integral which appears above L0k l y gm l k cm 0 In order to obtain an approximation to this term we take a second order Taylor series expansion of the log likelihood about 0k Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Derivation of SIC We have A A 8 L nL0kiY z nL0kiy0k0k w 1 A 82nL kiy A 0 0 0 70 2k k 80mg k k 1 A I A A inL0kiY 0k70k njwhm gkiekx where 2 A A 18 L0 10k7y77 w 7 80k80k is the average observed Fisher information matrix Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Derivation of SIC 0 Thus A 1 A A A ka M e Lm ly exp 75 or a or in mm or a on a We therefore have the following approximation for our integral Wk l y gm l to cm x Li kiy expe eke ki nii mi arm gwkik dok Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection v SIC Derivation D Derivation of SIC o The preceding Taylor series approximation holds when 0k is close to 0k 0 Thus the approximation of our integral should be valid for large n o In this instance L0kly should dominate the prior gt9lt l k within a small neighborhood of 0k 0 Outside of this neighborhood L09lt ly and the exponential term should be small enough to force the corresponding integrands near zero Joe Cavanaugh 17129 Advanced Bioslali cs Seminar Model Selection Derivation of SIC 0 Now consider evaluating the integral on the right hand side of Liokiy gm l to cm x A 1 A i A A L0k ly exp 75 0k 7 0k in mm or a an g0klk Wk using the noninformative prior g0k l k 1 o In this case we have exp 7 0k 7 ky n i ky 0k 7 dek MW in i kyl 12 Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Derivation of SIC 0 We therefore have L0kYg0kk d0k Hwy MM2 n j kyy l2 Hwy MM2 NH myN Iz A 2 kQ A Lwy Emmi2 22 Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Derivation of SIC o The preceding can be viewed as a variation on the Laplace method of approximating the integral Wk l y gm l to clot See Kass and Raftery 1995 o This approximation is valid so long as gt9lt l k is noninformative or flat over the neighborhood of 0k where L09lt ly is dominant although the choice of g0k l k 1 makes our derivation more tractable Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Derivation of SIC 0 We can now write 5ky 721mm 72InL0ky gwk cm 721mm kQ 72 In Ma y m Tz 721mm 72 In L ky kln In mam 17129 Advanced Bioslalislics Seminar Model Selection Joe Cavanaugh v SIC Derivation D Derivation of SIC o Ignoring terms in the preceding that are bounded as the sample size grows to infinity we obtain kly zi2lnL0Aklyknn a With this motivation the Schwarz information criterion is defined as follows SIC 72nL klyknn 72lnfyl0k kln n Joe Cavanaugh 17129 Advanced Bioslali cs Seminar Model Selection ll Del39ivntiun Discussion SIC and Bayes Factors 0 Consider two candidate models MRI and Mk2 in a Bayesian analysis To choose between these models a Bayes factor is often used a The Bayes factor Bu is defined as a ratio of the posterior odds of MRI PUlt1 lYPk2 M7 to the prior odds of MRI 7rk17rk2 o If Bu gt1 model MRI is favored by the data if Bu lt1 model Mk2 is favored by the data 0 Kass and Raftery 1995 write The Bayes factor is a summary of the evidence provided by the data in favor of one scientific theory represented by a statistical model as opposed to another Joe Cavanaugh 17129 Advanced BioslaLislics Seminar Model Selection ll D9I39IWICIDVI Discussion SIC and Bayes Factors 0 Let SICk1 denote SIC for model MRI and let SICk2 denote SIC for model Mk2 Kass and Raftery 1995 argue that as naoo 72 In Bu 7 7 0 2 In 812 A Thus SICk1 7 SICk2 can be viewed as a rough approximation to 72 In Bu Kass and Raftery 1995 write The Schwarz criterion or BIC gives a rough approximation to 72 the logarithm of the Bayes factor which is easy to use and does not require evaluation of prior distributions It is well suited for summarizing results in scientific communication Joe Cavanaugh 17129 Advanced BioslaLislics Seminar Model Selection SIC and Bayes Factors o The use of SIC seemsjustifiable for model screening in largesample Bayesian analyses 0 However SIC is often employed in frequentist analyses 0 Some frequentist practitioners prefer SIC to AIC since SIC tends to choose fitted models that are more parsimonious than those favored by AIC 0 However given the Bayesian justification of SIC is the use of the criterion in frequentist analyses defensible Joe Cavanaugh 17129 Advanced Bio 39 Model Selection 39 EYE ZQQ x m3m ediE e lgg kam mew m del k ee m Lectwe EX Cmi tuem ii r Time M ccillt l gamma Pam it Joe Cava naugh Department of Biostatistics College of Public Health The University of Iowa April 5 2005 Joe Cavanaugh mmgmmms witnein I F Lecture IX Criteria for Time Series Model Selection Part I 0 Brief Introduction to Time Series Analysis in Stationarity 0 Time and Frequency Domains 0 Autoregressive Models a Moving Average Models 0 Autoregressive Model Selection Framework 0 Final Prediction Error FPE ll Advanced Bio aLi 39 39 Model Selection Introduction FrameWi Introduction to Time Series 0 We will assume that the data vector y consists of n measurements on a response variable collected over equally spaced time points indexed by t 127 7 n 0 We will denote the response measurements as y17y27 7yn o In time series applications we assume that the yt are temporally correlated 0 Time series methodologies attempt to both characterize and utilize this temporal correlation 0 Most time series methodologies can be classified as belonging to either the time domain or the frequency domain ll Advanced Bio aLi 39 39 Model Selection Introduction Frame duction to Time Series Four important tools in time series analysis are the mean function the variance function the autocovariance function and the autocorrelation function ACF 0 Mean function W E Eyt a Variance function a E Varyt o Autocovariance function Cytmyt E Covytmyt o Autocorrelation function ACF Rytm7yt E Cyfi myt 2 2 Utmat Joe Cavanaugh 17129 Advanced Bio 39 Model Selection Introduction Frame Stationarity o A common assumption utilized in time series analysis is that of weak stationarity o The series yt is weakly stationary if the mean function W is constant M E In and ii the autocovariance function CyHrmyt depends only on the lag m Cytmyt E Cm a As a consequence of ii the variance function a is constant a CO E 02 0 Under stationarity the autocorrelation function can be written as Rm Cm02 Joe Cavanaugh 17129 Advanced Bio 39 Model Selection Introduction Stationarity o The following time series plot displays weekly mortality counts in Los Angeles county during the 197039s from January 1 1970 to December 1 1979 a Does the series appear stationary Joe Cavanaugh 17129 Advanced Bioslali cs Seminar Model Selection Introduction F1Irmwwiz FPE Stationarity gt o z x x x x x x x x x x o so 1oo 15o zoo 25o 3oo 35o A00 A50 soo Week Index Joe Cavanmmh 171 29 Advanced Model Select on Introduction Frame Time Domain Methodologies 0 Time domain methodologies directly analyze and model the original sample y1y2 7y Popular modeling frameworks autoregressive AR moving average MA autoregressive moving average ARMA autoregressive integrated moving average ARIMA autoregressive conditionally heteroscedastic ARCH generalized autoregressive conditionally heteroscedastic GARCH state space or dynamic linear modeling framework Common challenge To realistically model the autocovariance function a Common objective forecasting or prediction Joe Cavanaugh 17129 Advanced Bio 39 Model Selection Introduction Frame omain Methodologies 0 Frequency domain methodologies transform the original sample y17y27 7yr using a discrete Fourier transform DFT o The transformed data is indexed by a frequency 1 as opposed to a time t o The central objective is to characterize the frequencies and periodicities in the series 0 Nonstationary series can be analyzed in the frequency domain using a either a localized DFT or a discrete wavelet transform DWT Joe Cavanaugh 17129 Advanced Bio 39 Model Selection Introduction Fr Autoregressive Model 0 An autoregressive process of order p ARp is defined as yt 1y1371 2yt72 pyt7p en where et N iid N0702 o The autoregressive coefficients gt17 gt2 gtp must satisfy certain conditions for the ARp process to be stationary o The ACF for an ARp process decays quickly but is nonzero for all lags 0 Common problem in AR modeling the determination of the order p Joe Cavanaugh 17129 Advanced Bioslali cs Seminar Model Selection Introduction Framework FPE Moving Average Model 0 An moving average process of order q MAq is defined as Yr er 0mm 0M4 oqe h where et N iid N0702 o For any values of the moving average coefficients 0102 70 the MAq process is stationary o The ACF for an MAq process decays until lag q and is zero for all lags beyond q 0 Common problem in MA modeling the determination of the order q Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Introduction Frame AR and MA Relationships 0 An invertible MAq process can be represented as an infinite order autoregression ARoo with coefficients 1 that decay in magnitude as 139 increases 0 The moving average coefficients 0102 70q must satisfy certain conditions for the MAq process to be invertible o A stationary ARp process can be represented as an infinite order moving average MAoo with coefficients 0 that decay in magnitude as 139 increases Joe Cavanaugh 17129 Advanced Bio 39 Model Selection Innu n Framework FPE Autoregressive Model Selection Framework 0 True or generating model fyl0o a Candidate or approximating model fyl0k o Candidate class flk VOW l0k 6 9W 0 Assume fyl 0k corresponds to an autoregressive model of order p Note that k p l 1 0 Parameter vector 0k gt17 gt2 gtp702l 0 True parameter vector 00 gtf gtg 207 05 o Fitted model fyl k Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection In rev ion Framework Fl E Popular Criteria for Autoregressive Model Selection 0 The Akaike 1973 information criterion AIC 72 In fyl Q 2p 1 0 The corrected Akaike 1973 information criterion Hurvich and Tsai 1989 A 2 1 n AlCc 72 ln fyl 0k o The Schwarz 1978 information criterion SIC 72 In fyl k p 1 ln n o Other popular criteria for autoregressive model selection final prediction error FPE Akaike 1969 the Hannah and Quinn 1979 criterion HQ Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Final Prediction Error FPE 0 Final prediction error FPE was proposed by Akaike 1969 for the selection of the order of an autoregression o In the large samplejustification of FPE we assume that fyl 00 E fk as in the largesamplejustification of AIC o In the autoregressive setting this assumption amounts to requiring that the order of the true autoregressive model p0 is less than or equal to p the order of the candidate autoregressive model Joe Cavanaugh 17129 Advanced Bioslali cs Seminar Model Selection In ewm39i FP E Final Prediction Error FPE 0 Suppose we wish to choose a fitted ARp model that will yield an accurate predictor of yn1 o For an ARp model Yt 1Yt71 ZYt72 th7p 9157 the mean square error of prediction MSEP for forecasting yn1 is given by 109000 E Yn1 gt1yn ZYn71 pyn7p12 Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Fral n ewm39i FP E Final Prediction Error FPE o The mean square error of prediction MSEP can be viewed as a discrepancy ie a measure that reflects the disparity between the true model fyl 00 and the candidate model lel 0k 0 MSEP depends upon both the parameters of the true model 00 and the parameters of the candidate model 0k 0 Let Rom denote the true ACF One can show that 1090 0k a a gti 7 gtRo j gtj gtf7 p 1 11 j where O 7 O 7 7 0 7 po1 po2 p 0 Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection In ewm39i FP E Final Prediction Error FPE o The corresponding expected discrepancy is given by A090 k Ed0o 6 USUSE 2 p 11 j 43 7 gtRoi J39Xa j lt15 1 o In largesample settings p p A A nZZwi a gtRoi ijll j e gtf i1 j1 has an approximate chi squared distribution with p degrees of freedom Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection Final Prediction Error FPE 0 Thus if n is large we have Awayk Joe Cavanaugh 17129 Advanced Bioslalislics Seminar Model Selection

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